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Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation

Published online by Cambridge University Press:  28 May 2015

Zichao Di ,
Maria Emelianenko  and
Stephen Nash
Zichao Di*
Affiliation:
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
Maria Emelianenko*
Affiliation:
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
Stephen Nash*
Affiliation:
Systems Engineering and Operations Research Department, George Mason University, Fairfax, VA 22030, USA
*
Corresponding author.Email address:zdi@gmu.edu
Corresponding author.Email address:memelian@gmu.edu
Corresponding author.Email address:snash@gmu.edu
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Abstract

In a variety of modern applications there arises a need to tessellate the domain into representative regions, called Voronoi cells. A particular type of such tessellations, called centroidal Voronoi tessellations or CVTs, are in big demand due to their optimality properties important for many applications. The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings. This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems. Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems, and O(k) complexity estimation is provided for a problem with k generators.

Keywords

15A1265F1065F15Centroidal Voronoi tessellationoptimal quantizationtruncated Newton methodLloyd’s algorithmmultilevel methoduniform convergence
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 2 , May 2012 , pp. 242 - 259
Copyright
Copyright © Global Science Press Limited 2012

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